Upper join-closed subgroup property
This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
View a complete list of subgroup metaproperties
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions
This article is about a general term. A list of important particular cases (instances) is available at Category:Upper join-closed subgroup properties
Contents
Definition
Definition with symbols
A subgroup property is said to be upper join-closed if given
and
are intermediate subgroups of
containing
(indexed by a nonempty set
) and
satisfies
in each
, we have that
satisfies
in the join of subgroups
.
Relation with other metaproperties
Stronger metaproperties
- Lower-intersection upper-join closed subgroup property
- LU-join closed subgroup property
- Upward-closed subgroup property
- Izable subgroup property
Weaker metaproperties
Related notions
Given a subgroup property that is identity-true, upper join-closed and also satisfies the intermediate subgroup condition, we can, given any subgroup
of
associate a unique largest subgroup
containing
for which
satisfies
in
.
Such a subgroup property is termed an izable subgroup property and the that we get is termed the izing subgroup of
for that subgroup property.
Properties satisfying it
Normality
Normality is an upper join-closed subgroup property, viz, if and
are intermediate subgroups such that
and
, then
.
Central factor
The property of being a central factor is also upper join-closed, in fact, it is izable.